In this paper, we first give a sharp variational characterization to the smallest positive constant $C_{VGN}$ in the following Variant Gagliardo-Nirenberg interpolation inequality:
$$
\int_{\mathbb{R}^N\times\mathbb{R}^N}{{|u(x)|^p|u(y)|^p}\over{|x-y|^\alpha}}dxdy\leq
C_{VGN} \|\nabla u\|_{L^2}^{N(p-2)+\alpha} \|u\|_{L^2}^{2p-(N(p-2)+\alpha)},
$$
where $u\in W^{1,2}(\mathbb{R}^N)$ and $N\geq 1$. Then we use this characterization to determine the sharp threshold of $\|\varphi_0\|_{L^2}$ such that the solution of $i\varphi_t = - \triangle \varphi + |x|^2\varphi -
\varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$ with initial condition $\varphi(0, x) = \varphi_0$ exists globally or blows up
in a finite time. We also outline some results on the applications of $C_{VGN}$ to the Cauchy problem of
$i\varphi_t = - \triangle \varphi -
\varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.